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Chapter 7: Q. 11 (page 614)

Explain why all the terms of a divergent geometric series are nonzero.

Short Answer

Expert verified

The terms of a divergent geometric series are nonzero because if any one of the term is zero, then the series will be convergent.

Step by step solution

01

Step 1. Given information

If the common ratio of a geometric progression is greater then one then the series is divergent geometric series.

02

Step 2. Explanation

Consider the divergent geometric series $\sum_{k = 1}^{\infty} c r^{k}, r > 1$ .

The geometric series has zero terms if either $c = 0, o r r = 0$ .

If any one of them is zero then the product $c r^{k}$ is zero.

03

Step 3. Convergent and divergent geometric series

The product $c r^{k}$ is zero, and then the geometric series $\sum_{k = 1}^{\infty} c r^{k}$ has the partial sum equal to zero.

The sequence of partial sum of the series is zero.

The series with zero is constant and is convergent.

Therefore, for divergent geometric series both c and r should be non-zero.

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Most popular questions from this chapter

Prove that if $\sum_{k = 1}^{\infty} a_{k}$ converges to L and $\sum_{k = 1}^{\infty} b_{k}$ converges to M , then the series $\sum_{k = 1}^{\infty} (a_{k} + b_{k}) = L + M$ .

Given a series $\sum_{k = 1}^{\infty} a_{k}$ , in general the divergence test is inconclusive when $a_{k} \to 0$ . For a geometric series, however, if the limit of the terms of the series is zero, the series converges. Explain why.

Find the values of x for which the series $\sum_{k = 0}^{\infty} {(\frac{3}{x})}^{k}$ converges.

Which p-series converge and which diverge?

True/False:

Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: If $a_{k} \to 0$ , then $\sum_{k = 1}^{\infty} a_{k}$ converges.

(b) True or False: If $\sum_{k = 1}^{\infty} a_{k}$ converges, then $a_{k} \to 0$ .

(c) True or False: The improper integral $\int_{1}^{\infty} f (x) d x$ converges if and only if the series $\sum_{k = 1}^{\infty} f (k)$ converges.

(d) True or False: The harmonic series converges.

(e) True or False: If $p > 1$ , the series $\sum_{k = 1}^{\infty} k^{- p}$ converges.

(f) True or False: If $f (x) \to 0$ as $x \to \infty$ , then $\sum_{k = 1}^{\infty} f (k)$ converges.

(g) True or False: If $\sum_{k = 1}^{\infty} f (k)$ converges, then $f (x) \to 0$ as $x \to \infty$ .

(h) True or False: If $\sum_{k = 1}^{\infty} a_{k} = L$ and ${S_{n}}$ is the sequence of partial sums for the series, then the sequence of remainders ${L - S_{n}}$ converges to $0$ .

See all solutions

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